For sparse model, the sparsity of \(\theta _0\)  is directly controlled through the size of active set of \(\theta _0\) . In terms of the sparsity of \(\Sigma \) , we follow [1]} to consider a family of sparse covariance matrix as follows:
\(\mathcal {F}_{\alpha }=\bigg \lbrace  \Sigma :\max _j\sum _{i}\lbrace  |\sigma _{ij}|:|i-j|>k \rbrace  \le Mk^{-\alpha }\;\forall k,\lambda _{\max }(\Sigma )\le M_0,\;\lambda _{\min }(\Sigma )\ge m_0>0 \bigg \rbrace .\) 
